# Motivation for Hua's identity

I ran into Hua's identity without intending to, meaning that I do not have a concrete reference available, and my background is not in Ring Theory.

It is apparent to me that the identity is something of a big deal, but I couldn't find any explanation why. Authors pretty much assume that if you are reading about it, you already have a feeling for how important it is. Can you give me a hint?

• I am not an expert, by I guess it is related to or motivated by the question of deciding when, given two rings R and S, a spectrum-preserving bijection must be a Jordan isomorphism – Yemon Choi Sep 10 '17 at 2:45
• Have you read the wiki page en.wikipedia.org/wiki/Hua%27s_identity ? An application (as well as its background in geometry) is given there. – WhatsUp Sep 10 '17 at 5:41
• @WhatsUp: Sure, that was my first stop, but read the explanation with the eyes of an outsider. It is not clear how the identity is relevant to the proof of Hua's theorem, or why the theorem is non-trivial, or what is its connection in turn to the FTPG... – Rodrigo A. Pérez Sep 10 '17 at 12:11
• Thanks... Not much of a helpful comment... – Rodrigo A. Pérez Sep 10 '17 at 17:43
• @RodrigoA.Pérez You may or may not be interested in a question where I asked about the motivation of Hua's identity. I did not feel like I got a satisfying answer there, and I would really like to see good answers to your question. – rschwieb Sep 11 '17 at 13:27

Hua's identity is used to prove that any additive map of a division ring into itself preserving inverses must be an automorphism or antiautomorphism. His identity puts the Jordan triple product $$aba$$ in terms of additions and inverses, hence showing that those maps are also Jordan automorphisms; but Jordan isomorphisms between simple algebras had been shown by Ancochea and Kaplansky (1947) to be either automorphisms or antiautomorphisms, and the result for general division rings was proved by Hua himself (1949), completing the proof for additive maps preserving inverses. The study of contexts in which Jordan homomorphisms $$f:R\rightarrow S$$ for $$R,S$$ associative rings can be described in terms of (associative) homomorphisms and antihomomorphisms has a long and rich history (e.g., Jacobson and Rickart for $$R$$ a matrix ring, Bresar for $$S$$ a semiprime ring).

1. Le théorème de von Staudt en géométrie projective quaternionienne (1942). Ancochea.

2. On semi-automorphisms of division algebras (1947). Ancochea.

3. Semi-automorphisms of rings (1947). Kaplansky.

4. On the automorphisms of a sﬁeld (1949). Hua.

5. Jordan homomorphisms of rings (1950). Jacobson, Rickart.

6. Jordan mappings of semiprime rings I, II (1989-91). Bresar.

Either this answer to my own question encapsulates the reason that Hua's identity is a big deal, or is a trivial consequence of the true raison d'être for the identity. I do not know which...

The video proves a version of the identity formatted so that the product $aba$ is expressed exclusively in terms of inverses and sums: $$aba = \left[ (a-b^{-1})^{-1}-a^{-1} \right]^{-1}+a$$ Is that it?

• Thanks @rschwieb! I have been family-busy, but would be interested to have a more in-depth conversation... – Rodrigo A. Pérez Oct 6 '17 at 5:48